PN diode example (1D)

This example builds on the 1D resistor example, showing how to create a floods structure with a doping profile that varies with x. After solving, the example compares floods's numerical results with an analytical solution.

New Concepts

• refining the grid with non-tagged lines
• Doping=f(x) : use "sel z="
• "sel z=" if-then-else syntax (?:)
• tcl "foreach" and "array"

1D PN Diode Deck

Create 1D structure - explanation

#Grid
line x loc=0.0 spac=0.1 tag=Top
line x loc=0.5 spac=0.01
line x loc=1.0 spac=0.1 tag=Bottom

mater add name=Silicon
region Silicon xlo=Top xhi=Bottom
init

#Contacts
contact name=VSS Silicon xlo=-0.1 xhi=0.1 add
contact name=GND Silicon xlo=0.9  xhi=1.1 add

Declare solution variables

DevicePackage
solution add name=DevPsi pde solve negative damp
solution add name=Elec   pde solve !negative
solution add name=Hole   pde solve !negative

Define constants

set T 300.0
set k 1.38066e-23
set q 1.619e-19
set Vt [expr {$k*$T/$q}]
set ni 1.1e10
set esi [expr 11.8 * 8.85418e-14]
set eps [expr $esi / $q]
set Emob 350.0
set Hmob 150.0
set small 1.0e-10

Ionized dopant profile - explanation

sel z=1.0e20*(x<0.5) name=Nd
sel z=1.0e17 name=Na
sel z=(Nd-Na) name=Doping

#plot doping
sel z=Doping
plot.1d symb=1

Bulk Equations

set eqnP "$eps * grad(DevPsi) + Doping - Elec + Hole"
set eqnE "ddt(Elec) - ($Emob) * $Vt * sgrad(Elec, DevPsi/$Vt)"
set eqnH "ddt(Hole) - ($Hmob) * $Vt * sgrad(Hole, -DevPsi/$Vt)"
pdbSetString Silicon DevPsi Equation $eqnP
pdbSetString Silicon Elec   Equation $eqnE
pdbSetString Silicon Hole   Equation $eqnH 

pdbSetDouble Silicon DevPsi DampValue $Vt
pdbSetDouble Silicon DevPsi Abs.Error 1.0e-9
pdbSetDouble Silicon Elec   Abs.Error 1.0e-5
pdbSetDouble Silicon Hole   Abs.Error 1.0e-5

Contact Equations

proc OhmicContact {Contact} {
    global Vt ni
    pdbSetBoolean $Contact Elec Flux 1
    pdbSetBoolean $Contact Hole Flux 1
    pdbSetBoolean $Contact DevPsi Flux 1
    pdbSetBoolean $Contact Elec Fixed 1
    pdbSetBoolean $Contact Hole Fixed 1
    pdbSetBoolean $Contact DevPsi Fixed 1
    pdbSetDouble $Contact Elec Flux.Scale 1.619e-19
    pdbSetDouble $Contact Hole Flux.Scale 1.619e-19
    pdbSetString $Contact DevPsi Equation "Nd - Na - Elec + Hole"
    pdbSetString $Contact Elec Equation "DevPsi - $Vt*log((Elec)/$ni) -$Contact"
    pdbSetString $Contact Hole Equation "DevPsi + $Vt*log((Hole)/$ni) -$Contact"
}
OhmicContact VSS
OhmicContact GND

Initial Conditions - explanation

#Bias contacts
contact name=VSS voltage supply=0.0
contact name=GND voltage supply=0.0

#Initial Guess
sel z= {(Doping>0.0) 
            ?  ( 0.025*log( (Doping+$small) / $ni))
            :  (-0.025*log(-(Doping+$small) / $ni))} name = DevPsi
sel z=$ni*exp(DevPsi/$Vt)  name=Elec
sel z=$ni*exp(-DevPsi/$Vt) name=Hole

1st DC Solve at Equilibrium (0V)

device
puts "Electron Flux [contact name=VSS sol=Elec flux]"
puts "Hole Flux [contact name=VSS sol=Hole flux]"

#Plot the equilibrium concentration profiles
foreach var {Doping Elec Hole} {
    sel z=log10(abs($var+1.0))
    plot.1d !cle
}

Ramp the DC Bias to make I-V plot

#initialize an array, bias (Vb) and counter (i), and create and new graph window
array set curr {}
#set Vb 0.0
set i 1

#Ramp from 0.0-1.2V and Plot
set Win [CreateGraphWindow]
for {set bias 0.0} {$bias < 1.2} {set bias [expr $bias+0.05]} {
   set Vbias($i) -$bias
   incr i

   contact name=VSS supply=-$bias
   device init

   set curr($i) [expr ([contact name=VSS sol=Elec flux] - [contact name=VSS sol=Hole flux])]

   sel z=log10(abs(Hole)+1.0)
   plot.1d !cle

   AddtoLine $Win IV -$bias $curr($i)
}

#Get the last few points on the IV curve to use for linear extrapolation of the built-in voltage
set l [list "$Vbias(24)" "$curr(24)" "$Vbias(23)" "$curr(23)" "$Vbias(22)" "$curr(22)" ]

#Fit the data points with "Bestline". The output will give the x-intercept or built-in voltage
Bestline $l

#Create a "fit" line to plot on top of IV curve using the data points and "Bestline" x-intercept result
set line [list "$Vbias(24)" "$curr(24)" "$Vbias(23)" "$curr(23)" "$Vbias(22)" "$curr(22)" 0.9998 0] 
CreateSingleLine  $Win Vbi_fit $line

In forward bias (negative voltage applied to the top (n-type) contact), the output IV plot with the absolute value of the built-in voltage equal to 0.9998 is shown below. The built in voltage calculated by the equation Vbi=Vt*ln((Nd*Na)/ni^2) is 0.9786.

Full Deck Without Explanations

Copy-paste this entire deck into a file (for example, 1dpndiode.tcl) to make running it easy. Use the startup script to alias your paths to the floods executable. Then, on the BASH and then flooxs command line type:

$ floods
flooxs> source 1dpndiode.tcl